I've tried playing around with a hybrid opening recently that tries to generate a large amount of culture to start. It relies on getting Stonehenge and I've only gotten up to Emperor, so I'm not sure it would fly on higher difficulty.
It usually also requires having two luxuries somewhere near the capital to sell to other civs for one-time lumps of gold.
It goes: Tradition (3 culture in capital) -> Liberty (1 additional culture) -> Citizenship (free worker) -> Representation (less policy penalty when founding new cities) -> Collective Rule (free settler) -> Meritocracy (free engineer) -> Legalism (free temples)
If there's enough gold generated at the start and I can often sell a luxury for 300 gold, then a cultural city-state can be bought off for 500 gold which boosts the process along.
I've started building Monument -> Scout -> Warrior -> Warrior/Scout -> Stonehenge. It leaves the terrain at the capital undeveloped until Citizenship comes around, but it's good for early defense.
After Stonehenge, work starts on a Settler. Within a few turns of the Settler finishing production, I can usually sell my second luxury to work towards buying a 2nd Settler. Collective Rule provides a 3rd settler, and with the 3 settlers I can get out 4 early cities (though the other civs have normally each founded a 2nd city by this point).
Depending on the terrain, I've built monuments in the other 3 cities pretty quickly, though sometimes squeezed them in just before Legalism popped. I forget which tech provides temples, but it can be grabbed in enough time to get 4 free temples for all the starting cities.
I've been using the Great Engineer from meritocracy to pop the Collossus in a seaside city for later gold generation. I haven't tried popping the Hanging Gardens, but that could provide a large early boost as well. The Great Engineer can also pop the Oracle, and if the Monuments are down early enough then Legalism could be grabbed from that. Landed Elite or Monarchy could then be grabbed off the next policy, depending on whether you want to grow the frontier cities or the capital.
I've used this in my last three games, but only on Emperor. It is terrain dependant, relying on the early nearby luxuries (in the capital and at frontier cities). Also, it helps to be able to set up highly defensible frontier cities with a few archers built for defense. It's been pretty fun starting awash in culture, though it does hurt the tech pace a little.